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Properties

Label 1309.cn
Modulus $1309$
Conductor $1309$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1309, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([10,24,45])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(38,1309)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1309\)
Conductor: \(1309\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(60\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(12\) \(13\)
\(\chi_{1309}(38,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{1309}(47,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{1309}(115,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{1309}(157,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{1309}(234,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{1309}(565,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{1309}(642,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{1309}(752,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{1309}(829,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{1309}(922,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{1309}(999,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{1309}(1109,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{1309}(1160,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{1309}(1186,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{1309}(1237,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{1309}(1279,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{3}{10}\right)\)